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Gregory M. Johnson - One of the best experts on this subject based on the ideXlab platform.

  • ABSTRACT Title of dissertation: Abstract Elementary Classes With Löwenheim-Skolem Number
    2008
    Co-Authors: Cofinal Ω, Gregory M. Johnson
    Abstract:

    An abstract Elementary class is a class K of structures for some vocabulary L together with a “strong Substructure ” relation ≺K on K satisfying certain axioms. Abstract Elementary classes include Elementary classes with Elementary Substructure and classes axiomatizable in L∞,ω with Elementary Substructure relative to some fragment of L∞,ω. For every abstract Elementary class there is some number κ, called the Löwenheim-Skolem number, so that every structure in the class has a strong Substructure of cardinality ≤ κ. We study abstract Elementary classes with Löwenheim-Skolem number κ, where κ is cofinal with ω, which have finite character. We generalize results obtained by Kueker for κ = ω. In particular we show that K is closed under L∞,κ-Elementary equivalence and obtain sufficient conditions for K to be L∞,κ-axiomatizable. The results depend on developing an appropriate concept of κ-a.e. Abstract Elementary Classes with Löwenheim-Skolem Number Cofinal with ω b

Azadeh Neman - One of the best experts on this subject based on the ideXlab platform.

  • Embeddings and chains of free groups
    2008
    Co-Authors: Eric Jaligot, Azadeh Neman
    Abstract:

    We build two nonabelian CSA-groups in which maximal abelian sub-groups are conjugate and divisible, as the countable unions of increasing chains of CSA-groups and by keeping the constructions as free as possible in each case. For n ≥ 1, let Fn denote the free group on n generators. We view all groups G as first-order structures 〈G, ·, −1, 1〉, where ·, −1, and 1 denote respectively the multiplication, the inverse, and the identity of the group. The following striking results are proved in a series of papers of Sela culminating in [Sel07]. Fact 1 [Sel05, Sel06a, Sel06b, Sel07] (1) For any 2 ≤ n ≤ m, the natural embedding Fn ≤ Fm is an Elementary embedding. (2) For any n ≥ 2, the (common) complete theory Th (Fn) is stable. We refer to [Hod93] for model theory in general, and to [Poi87] and [Wag97] for stability theory and in particular stable groups. Let F denote the free group over countably many generators. Fact 1 has the following corollary. Corollary 2 The natural embeddings F2 ≤ · · ·Fn ≤ · · · ≤ F are all Elementary. In particular each Fn is an Elementary Substructure of F, and Th (F) is stable. A CSA-group is a group in which maximal abelian subgroups A are mal-normal, i.e., such that A ∩ Ag 6 = 1 implies that g is in A for any element g of the ambient group. The class of CSA-groups contains all free groups and is studied from various points of view. We refer to [JOH04, JMN08] for a model theoretic approach in combination of questions concerning particular groups [Che79, Jal01, Cor03], and to [KMRS08] for an approach more related to com-putational aspects in limit groups. We prove the following lemma on embeddings of torsion-free CSA-groups in which maximal abelian subgroups are cyclic. 1 ha l-

  • Embeddings and chains of free groups
    2008
    Co-Authors: Eric Jaligot, Azadeh Neman
    Abstract:

    We build a nonabelian CSA-group in which maximal abelian subgroups are conjugate and divisible and in which quantifier-free definable sets are stable. For n ≥ 1, let Fn denote the free group on n generators. We view all groups G as first-order structures 〈G, ·, −1, 1〉, where ·, −1, and 1 denote respectively the multiplication, the inverse, and the identity of the group. The following striking results are proved in a series of papers of Sela culminating in [Sel07]. Fact 1 [Sel05, Sel06a, Sel06b, Sel07] (1) For any 2 ≤ n ≤ m, the natural embedding Fn ≤ Fm is an Elementary embedding. (2) For any n ≥ 2, the (common) complete theory Th (Fn) is stable. We refer to [Hod93] for model theory in general, and to [Poi87] and [Wag97] for stability theory and in particular stable groups. Let F denote the free group over countably many generators. Fact 1 has the following corollary. Corollary 2 The natural embeddings F2 ≤ · · · Fn ≤ · · · ≤ F are all Elementary. In particular each Fn is an Elementary Substructure of F, and Th (F) is stable. A CSA-group is a group in which maximal abelian subgroups A are malnormal, i.e., such that A ∩ A g = 1 implies that g is in A for any element g of the ambient group. The class of CSA-groups contains all free groups and is studied from various point of view. We refer to [JOH04, JMN08] for a model theoretic approach in combination of questions concerning particular groups [Che79, Jal01, Cor03], and to [KMRS08] for an approach more related to computational aspects in limit groups. We prove the following lemma on embeddings of free groups. Lemma 3 For any k ≥ 2, there exists an embedding φ of F into itself in such a way that maximal abelian subgroups of φ(F) are conjugate and any element of φ(F) has a k-th root. 1 Proof: F has countably many maximal abelian subgroups, and countably many conjugacy classes of such maximal abelian subgroups, which can be enumerated by i < ω. For each such conjugacy class, fix a maximal (cyclic) abelian subgroup Ai, and inside Ai fix a generator ai. We define inductively on i an increasing family of supergroups Gi of F as follows. • G0 = F. • Gi+1 is the HNN-extension 〈Gi, ti | a t

Stevo Todorcevic - One of the best experts on this subject based on the ideXlab platform.

  • ANEWLÖWENHEIM-SKOLEM THEOREM
    2008
    Co-Authors: Matthew Foreman, Stevo Todorcevic
    Abstract:

    Abstract. This paper establishes a refinement of the classical Löwenheim-Skolem theorem. The main result shows that any first order structure has a countable Elementary Substructure with strong second order properties. Several consequences for Singular Cardinals Combinatorics are deduced from this. 1

Vasey Sebastien - One of the best experts on this subject based on the ideXlab platform.

  • Symmetry in abstract Elementary classes with amalgamation
    'Springer Science and Business Media LLC', 2017
    Co-Authors: Vandieren, Monica M., Vasey Sebastien
    Abstract:

    This paper is part of a program initiated by Saharon Shelah to extend the model theory of first order logic to the non-Elementary setting of abstract Elementary classes (AECs). An abstract Elementary class is a semantic generalization of the class of models of a complete first order theory with the Elementary Substructure relation. We examine the symmetry property of splitting (previously isolated by the first author) in AECs with amalgamation that satisfy a local definition of superstability. The key results are a downward transfer of symmetry and a deduction of symmetry from failure of the order property. These results are then used to prove several structural properties in categorical AECs, improving classical results of Shelah who focused on the special case of categoricity in a successor cardinal. We also study the interaction of symmetry with tameness, a locality property for Galois (orbital) types. We show that superstability and tameness together imply symmetry. This sharpens previous work of Boney and the second author.Comment: 37 pages. This merges with arXiv:1509.01488 . Was previously titled "Transferring symmetry downward and applications

Sebastien Vasey - One of the best experts on this subject based on the ideXlab platform.

  • Symmetry in abstract Elementary classes with amalgamation
    Archive for Mathematical Logic, 2017
    Co-Authors: Monica M. Vandieren, Sebastien Vasey
    Abstract:

    This paper is part of a program initiated by Saharon Shelah to extend the model theory of first order logic to the non-Elementary setting of abstract Elementary classes (AECs). An abstract Elementary class is a semantic generalization of the class of models of a complete first order theory with the Elementary Substructure relation. We examine the symmetry property of splitting (previously isolated by the first author) in AECs with amalgamation that satisfy a local definition of superstability. The key results are a downward transfer of symmetry and a deduction of symmetry from failure of the order property. These results are then used to prove several structural properties in categorical AECs, improving classical results of Shelah who focused on the special case of categoricity in a successor cardinal. We also study the interaction of symmetry with tameness, a locality property for Galois (orbital) types. We show that superstability and tameness together imply symmetry. This sharpens previous work of Boney and the second author.